A Domain-Theoretic Foundation for Imprecise Probability and Credal Sets
2026-04-10 • Logic in Computer Science
Logic in Computer Science
AI summaryⓘ
The authors create a mathematical setup to work with uncertain probabilities that come from unclear events or groups of possible probability values (called credal sets) on spaces wider than simple cases. They build ways to update these uncertain probabilities using Bayesian rules even when the uncertainty is complex, and also explore how to understand when events are conditionally independent in this unclear setting. Their work links logical reasoning, topology, and measure theory, making the approach both sound and practical. Additionally, they introduce new types of credal sets based on repeated functions with uncertain weights, making the models easier to compute.
domain theoryimprecise probabilitycredal setsBayesian updatingconditional independencetopological spacesScott continuityChoquet integrationiterated function systemscapacity theory
Authors
Abbas Edalat, Pietro Di Gianantonio, Amin Farjudian
Abstract
We develop a domain-theoretic framework for imprecise probability reasoning and inference on general topological spaces with a countably based continuous lattice of open sets. We address two distinct forms of uncertainty: partial or incomplete event descriptions, and sets of probability distributions as represented by credal sets -- as well as their combination. Within this framework, we construct a theory of conditional probability and derive novel inference rules for performing Bayesian updating in the presence of these two complementary types of imprecision. These results are extended to a theory of conditional independence for imprecise probabilistic events. We also formulate logical predicates for conditional probability, Bayesian updating, and conditional independence, and we obtain the relevant soundness and completeness results. A key contribution is the construction of a Scott-continuous mapping from any credal set to the domain of intervals, providing a domain-theoretic realisation of classical results from capacity theory and Choquet integration. Finally, we introduce and study a new family of credal sets generated by iterated function systems with imprecise probability weights, broadening the scope of computationally tractable imprecise probabilistic models. The resulting computable framework unifies logical, topological, and measure-theoretic perspectives on uncertainty, supporting robust probabilistic inference under partial and set-valued information.